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A121637 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns (n>=1; 0<=k<=n-1). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. 2
1, 1, 1, 2, 3, 1, 7, 10, 6, 1, 29, 47, 33, 10, 1, 147, 265, 210, 82, 15, 1, 889, 1740, 1521, 697, 171, 21, 1, 6252, 13087, 12373, 6377, 1885, 317, 28, 1, 50163, 111066, 112016, 63261, 21390, 4407, 540, 36, 1, 452356, 1050608, 1118991, 680541, 255245, 60903 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums are the factorials (A000142). T(n,0)=A121638(n). Sum(k*T(n,k), k=0..n-1)=A121639(n)

REFERENCES

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

LINKS

Table of n, a(n) for n=1..51.

FORMULA

The row generating polynomials are P(n,t)=Q(n,t,1,1), where Q(1,t,x,y)=x, Q(2,t,x,y)=x+ty and Q(n,t,x,y)=Q(n-1,t,ty,1/t)+(x+ty+n-3)Q(n-1,t,1,1) for n>=3.

EXAMPLE

T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 2-cell columns.

Triangle starts:

1;

1,1;

2,3,1;

7,10,6,1;

29,47,33,10,1;

MAPLE

Q[1]:=x: Q[2]:=x+t*y: for n from 3 to 11 do Q[n]:=sort(expand(subs({x=t*y, y=1/t}, Q[n-1])+(x+t*y+n-3)*subs({x=1, y=1}, Q[n-1]))) od: for n from 1 to 11 do P[n]:=sort(subs({x=1, y=1}, Q[n])) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form

CROSSREFS

Cf. A000142, A121638, A121639, A121554.

Sequence in context: A085588 A118008 A173459 * A247370 A161847 A101175

Adjacent sequences:  A121634 A121635 A121636 * A121638 A121639 A121640

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 13 2006

STATUS

approved

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Last modified May 18 12:09 EDT 2021. Contains 343995 sequences. (Running on oeis4.)